A high order numerical method for 2D time-fractional neutron diffusion equation characterizing neutron transport in nuclear reactors

In this article, we investigate a numerical method for solving a two-dimensional time-fractional neutron diffusion equation that incorporates three Caputo fractional derivatives in time. First, the existence, uniqueness, and solution representation are established. We then develop an efficient numerical scheme using the linear Galerkin finite element method for spatial discretization and a second-order convolution quadrature for temporal discretization. Error estimates for both semidiscrete and fully discrete schemes are derived. Our proposed numerical method is demonstrated to achieve second-order convergence in both space and time. Numerical experiments are conducted to validate the theoretical results. Neutron fluxes are computed for both short and long simulation times to analyze the behavior of neutron diffusion under varying orders of the fractional derivative. Additionally, a comparison is made with the method presented in Roul et al. (2020) to highlight the advantages of our proposed approach.